特征值与图的结构

项目名称： 特征值与图的结构

项目编号： No.11201198

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李红海

作者单位： 江西师范大学

项目金额： 22万元

中文摘要： 谱图理论主要研究图的代数表示（图对应的矩阵）的谱，通过讨论图的特征空间，建立图的拓扑结构与图的特征值之间的联系，应用代数理论、几何理论与概率方法来研究图的拓扑结构性质，以及应用图的拓扑结构来研究代数和几何中的谱性质。 本项目主要研究谱图理论中国际上重点关注的几个问题，包括图的规范化Laplacian与距离矩阵、图的零度及其在化学的应用。研究图的规范化Laplacian的次小特征值与谱半径，建立其与图的各种不变量之间的联系。研究图的规范化Laplacian的次小特征值与谱半径的谱扰动，由此给出各种图类中具有极端规范化Laplacian次小特征值与谱半径的极图刻画。研究规范化Laplacian次小特征值对应的调和特征函数所反映的图的组合结构性质。探索依秩或零度对二部图的分类问题，给出基本图类的零度刻画。研究图的距离矩阵的谱半径与行列式。

中文关键词： 图；超图；特征值；谱半径；匹配

英文摘要： Spectral graph theory is the study of the eigenvalues of the algebraic representations (i.e. matrices corresponding to graphs)of graphs. By investigating the eigenspace of graphs, set up the relation between eigenvalues and topological structure of graphs.Use algebraic theory, geometric theory and probabilistic method to study the structure properties of graphs and apply graph theory to study the spectral problem in algebra and geometry. The project is devoted to the following problems which have attracted much attention of international spectral researchers, including the normalized Laplacian and distance matrix of graphs, the nullity of graphs and their applications in chemistry. We shall study the second smallest eigenvalue and spectral radius of the normalized Laplacian and find the relation between them and various invariants of graphs. Meantime, we shall study the spectral perturbation on the second smallest eigenvalue and spectral radius of the normalized Laplacian of graphs, perhaps from which we can characterize the graphs with extremal value of the second smallest eigenvalue and spectral radius of the normalized Laplacian among various classes of graphs. Try to extract the information of the graph structure from the harmonic eigenfunction corresponding to the second smallest eigenvalue of the normali

英文关键词： graph；hypergraph；eigenvalue；spectral radius；matching